Notes & slides

These are (expanded) slides from a 15-minute talk at CNTA XIV discussing my (upcoming) paper on the equidistribution of shapes of complex cubic fields with fixed trace-zero form on the corresponding geodesics in the modular curve. (The animations don't work in Preview on Mac.)

This short note explains how to relate the critical integers of a motivic L-function (in the sense of Deligne) to the Hodge numbers of the motive. Specific examples are worked out (the Tate motive Q(1) and symmetric powers of modular forms).

Graphics

Some data visualizations I've created. Click on an image for a description of it.

Code

Overconvergent modular symbols code (including Hida theory):This code is available here on github. See the published article.

Iterate factorization trees:

This code is available here on github. It was first written at the AIM workshop on the Galois theory of orbits in arithmetic dynamics and computes the
trees described in Section 3 of Rafe Jones and Nigel Boston's Settled polynomials over finite fields.

Shape of a number field:

This code is available here on github. It computes the "shape" of a number field. The shape of a number field K of degree n is the (n−1)-dimensional lattice in the Minkowski space given by the orthogonal complement of the vector 1. The output of this function is simply the (not necessarily reduced) Gram matrix of the lattice.

Artin representations in Sage:

This code is available here on github. The goal of this project is to be able to compute with Artin representations in Sage. Some highlights include Artin conductors, L-functions, and root numbers. The project also improves the Galois group code to allow non-Galois fields.